Unbounded Algebraic Derivators

TL;DR

This paper develops a derivator framework for unbounded derived categories of Grothendieck categories, enabling new insights into localization and group cohomology over arbitrary rings.

Contribution

It introduces a derivator structure for unbounded derived categories and applies it to localization and group cohomology in a novel way.

Findings

Derived category of a Grothendieck category forms a derivator with all small categories as diagrams.

Provides a description of localization functors related to specialization closed subsets.

Proves basic theorems of group cohomology for complexes over arbitrary base rings.

Abstract

We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring.